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EXPONENTIAL DECAY of LEBESGUE NUMBERS Peng Sun 1. Motivation Entropy, Which Measures Complexity of a Dynamical System, Has Vario

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EXPONENTIAL DECAY of LEBESGUE NUMBERS Peng Sun 1. Motivation Entropy, Which Measures Complexity of a Dynamical System, Has Vario EXPONENTIAL DECAY OF LEBESGUE NUMBERS Peng Sun China Economics and Management Academy Central University of Finance and Economics No. 39 College South Road, Beijing, 100081, China Abstract. We study the exponential rate of decay of Lebesgue numbers of open covers in topological dynamical systems. We show that topological en- tropy is bounded by this rate multiplied by dimension. Some corollaries and examples are discussed. 1. Motivation Entropy, which measures complexity of a dynamical system, has various defini- tions in both topological and measure-theoretical contexts. Most of these definitions are closely related to each other. Given a partition on a measure space, the famous Shannon-McMillan-Breiman Theorem asserts that for almost every point the cell covering it, generated under dynamics, decays in measure with the asymptotic ex- ponential rate equal to the entropy. It is natural to consider analogous objects in topological dynamics. Instead of measurable partitions, the classical definition of topological entropy due to Adler, Konheim and McAndrew involves open covers, which also generate cells under dynamics. We would not like to speak of any in- variant measure as in many cases they may be scarce or pathologic, offering us very little information about the local geometric structure. Diameters of cells are also useless since usually the image of a cell may spread to the whole space. Finally we arrive at Lebesgue number. It measures how large a ball around every point is contained in some cell. It is a global characteristic but exhibits local facts, in which sense catches some idea of Shannon-McMillan-Breiman Theorem. We also notice that the results we obtained provides a good upper estimate of topological entropy which is computable with reasonable effort. 2. Preliminaries on Lebesgue number First we briefly discuss some preliminaries on Lebesgue number and open covers. Some of those can be found in any textbook of elementary topology. For the rest, as well as other facts we discuss in succeeding sections without proof, one can refer to, for example, [5]. The basic result we shall use is the following Lebesgue Covering Lemma. Theorem 2.1. Let (X, d) be a compact metric space. is an open cover of X. Then there is δ > 0 such that every open ball of radiusU at most δ is contained in some element of . We call the largest such number the Lebesgue number of the open cover. U 1991 Mathematics Subject Classification. Primary: 37B40, 54F45. Key words and phrases. Lebesgue number, dimension theory, topological entropy. 1 2 PENG SUN Proof. If X then the theorem is trivial. If X / ,∈U let ∈U δ( , x) = sup( inf d(x, y)). U U∈U y∈X\U Then δ( , x) is a continuous function on X taking strictly positive values. Since X is compact,U the function attains its minimum value on X. So δ( ) = min δ( , x) > 0 U x∈X U is the Lebesgue number of the open cover. Remark. Another widely used formulation of Lebesgue Covering Lemma states that there is δ¯ > 0 (the largest one) such that every set of diameter less than δ¯ is contained in some element of . It is easy to see δ δ¯ 2δ. This guarantees that Definition 3.1 is not affected ifU Lebesgue number is≤ defined≤ this way. We have some simple facts on Lebesgue numbers. Lemma 2.2. For two open covers and , we say is finer than , denoted by , if every element of is containedU inV some elementU of . If Vis finer than U≻Vthen δ( ) δ( ). U V U V U ≤ V Lemma 2.3. Let diam( ) = supU∈U diam(U). For every open covers and , if diam( ) <δ( ), then U . U V U V U≻V Lemma 2.4. For any two open covers and , let U V = U V U , V . U V { ∩ | ∈U ∈ V} Then _ δ( ) = min δ( ),δ( ) . U V { U V } Proof. On one hand, is finer_ than and . By Proposition 2.2 U V U V W δ( ) min δ( ),δ( ) U V ≤ { U V } On the other hand, for every x,_ there are U and V such that x ∈U x ∈V δ( , x) = min d(x, y) δ( ), δ( , x) = min d(x, y) δ( ) U y∈X\Ux ≥ U V y∈X\Vx ≥ V Then δ( , x) min d(x, y) = min δ( , x),δ( , x) min δ( ),δ( ) U V ≥ y∈X\(Ux∩Vx) { U V }≥ { U V } _ Now let f be a continuous map from X to itself. Let n−1 n = f −k( ), δ = δ (f, )= δ( n) Uf U n n U Uf k_=0 where f( )= f(U) U . U { | ∈U} Corollary 2.5. Let be an open cover, then U −k δn(f, ) = min δ(f ( )). U 0≤k≤n−1 U Corollary 2.6. If , then for every n, we have f −n( ) f −n( ) and n U ≻V U ≻ V Uf ≻ n, hence δ(f −n( )) δ(f −n( )) and δ (f, ) δ (f, ). Vf U ≤ V n U ≤ n V DECAY OF LEBESGUE NUMBERS 3 3. Decay of Lebesgue numbers Now we turn to the asymptotic decay of Lebesgue numbers. Definition 3.1. Let be an open cover of X. We set U − 1 h (f, ) = lim inf log δn(f, ), L U n→∞ −n U + 1 hL (f, ) = lim sup log δn(f, ), U n→∞ −n U h−(f) = sup h−(f, ) L L U and h+(f) = sup h+(f, ). L L U Here the supremums are taken over all finite open covers. ∗ + − From now on we use hL to denote either hL or hL , when the argument works for both cases. We note that these numbers possess some properties analogous to entropy. Lemma 3.2. For every continuous map f and every open cover , we have: ∗ ∗ U (1) hL(f, ) 0, hence hL(f) 0. U ≥ ∗≥ ∗ (2) If f is an isometry, then hL(f)= hL(f, )=0. − + − + U (3) hL (f, ) hL (f, ), hence hL (f) hL (f). (4) For everyU ≤n > 0, Uh∗ (f, ) = h∗ (f,f≤ −n( )) = h∗ (f, n). If in addition f L U L U L Uf is a homeomorphism, then the first equality also holds for n< 0. (5) If , then h∗ (f, ) h∗ (f, ). U≻V L U ≥ L V ∗ n ∗ Proposition 3.3. For every n> 0, hL(f )= nhL(f). Proof. On one hand, for every finite open cover and m> 0, by Corollary 2.5 U n −jn −j δm(f , ) = min δ(f ( )) min δ(f ( )) = δmn(f, ). U 0≤j≤m−1 U ≥ 0≤j≤mn−1 U U ∗ n ∗ ∗ n ∗ Taking limit we obtain hL(f , ) nhL(f, ), hence hL(f ) nhL(f). On the other hand, U ≤ U ≤ n n −jn n −j δm(f , f ) = min δ(f ( f )) = min δ(f ( )) = δmn(f, ). U 0≤j≤m−1 U 0≤j≤mn−1 U U This implies h∗ (f n, n) nh∗ (f, ). So h∗ (f n) nh∗ (f). L Uf ≥ L U L ≥ L Applying Corollary 2.5, we also have: Corollary 3.4. + 1 −n hL (f, ) lim sup log δ(f ( )). U ≥ n→∞ −n U 1 h−(f, ) lim inf log δ(f −n( )). L U ≥ n→∞ −n U (We intentionally replace f 1−n by f −n in the limits.) In fact, we have: Proposition 3.5. + 1 −n hL (f, ) = lim sup log δ(f ( )) U n→∞ −n U − Remark. The analogous result is not necessarily true for hL . 4 PENG SUN The proposition is a corollary of the following lemma. We also note that Lebesgue number is always bounded by the diameter of the space. Lemma 3.6. Let a K be real numbers, uniformly bounded from below. Let n ≥ b = max a 1 k n . n { k| ≤ ≤ } Then a b lim sup n = lim sup n n→∞ n n→∞ n Proof. By definition, a b . So n ≤ n a b lim sup n lim sup n . n→∞ n ≤ n→∞ n Note that b is a non-decreasing sequence. If there is N such that for all { n} n>N, bn = bN , then b K a lim sup n = 0 = lim lim sup n . n→∞ n n→∞ n ≤ n→∞ n Otherwise, the set J = n b <b has infinitely many elements. If n J { j| nj −1 nj } j ∈ then we must have anj = bnj . b bn an a lim sup n = lim sup j = lim sup j lim sup n . n→∞ n j→∞ nj j→∞ nj ≤ n→∞ n Proposition 3.7. ∗ ∗ hL(f) = lim inf hL(f, ). ǫ→0 diam(U)<ǫ U Proof. By definition, h∗ (f) h∗ (f, ) for every open cover . So L ≥ L U U ∗ ∗ hL(f) lim sup inf hL(f, ). ≥ ǫ→0 diam(U)<ǫ U ∗ ∗ For every θ> 0, if hL(f, ) >hL(f) θ, then for every ǫ<δ( ) and every open cover with diam( ) <ǫ, weV have h∗ (−f, ) h∗ (f, ) >h∗ (f)V θ. So U U L U ≥ L V f − ∗ ∗ hL(f) lim inf inf hL(f, ). ≤ ǫ→0 diam(U)<ǫ U Remark. This proposition implies that in Definition 3.1 the supremums may be taken over all open covers (not necessarily finite). Corollary 3.8. For every sequence of open covers k k≥1, if diam( k) 0, then h∗ (f, ) h∗ (f). {U } U → L Uk → L Corollary 3.9. If f is an expansive homeomorphism with expansive constant γ, then for every open cover of diameter less than γ, h∗ (f, )= h∗ (f). U L U L Proof.
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